Elliptical magnetic trajectories of a point on the elliptical 2-sphere

Abstract

Elliptical trajectories play numerous significant roles in mathematics and physics. The focus of this work is to analyze the trajectories of a point on the ellipsoid S a 1 , a 2 , a 3 2 while it is under the influence of a killing vector field K. For this purpose, we introduce the generalized Darboux frame and the variational vector fields of S a 1 , a 2 , a 3 2 . We were able to get the trajectory equations of the charged particle on the ellipsoid by utilizing the elliptical structure and elliptic inner and cross product. We adopted a clearer and more comprehensible strategy in this way. With the aid of a mathematical program, we were able to display the paths we had determined based on the particle's velocity vector's position in the magnetic field. These trajectories are spiral, helical, and elliptical.